Modeling Differences in the Dimensionality of Multiblock Data by Means of Clusterwise Simultaneous Component Analysis

Abstract

Given multivariate multiblock data (e.g., subjects nested in groups are measured on multiple variables), one may be interested in the nature and number of dimensions that underlie the variables, and in differences in dimensional structure across data blocks. To this end, clusterwise simultaneous component analysis (SCA) was proposed which simultaneously clusters blocks with a similar structure and performs an SCA per cluster. However, the number of components was restricted to be the same across clusters, which is often unrealistic. In this paper, this restriction is removed. The resulting challenges with respect to model estimation and selection are resolved.

Appendix:  Derivation of an AIC-Based Partition Criterion

Conditional upon a specific Clusterwise SCA-ECP model M, the log-likelihood of data block X _i when assigned to cluster k (and thus modeled by $\mathbf{M}_{i}^{(k)}$) amounts to

(A.1)

which is the block-specific counterpart of Equation (7), given $\mathit{SSE}_{i}^{(k)}$ as defined in Equation (5). When inserting $\hat{\sigma}^{2} = \frac{\mathit{SSE}_{i}^{(k)}}{N_{i} J}$ as a post-hoc estimator of the error variance σ ² (Wilderjans et al. 2011), the log-likelihood can be rewritten as

$$ \operatorname{loglik}\bigl(\mathbf{X}_{i}| \mathbf{M}_{i}^{(k)}\bigr) = - \frac{N_{i} J}{2}\bigl[ 1 + \log( 2\pi) - \log( N_{i }J ) + \log\bigl( \mathit{SSE}_{i}^{(k)} \bigr) \bigr], $$

(A.2)

where the first three terms are not influenced by the cluster assignment and can thus be discarded. The number of free parameters for data block i, when it is tentatively assigned to cluster k, is denoted by $\mathit{fp}_{i}^{(k)}$ and can be computed as follows:

$$ \mathit{fp}_{i}^{(k)} = N_{i}Q^{(k)}. $$

(A.3)

It corresponds to the size of the component score matrix $\mathbf{F}_{i}^{(k)}$ that is computed to evaluate the fit of data block i in cluster k. When combining Equations (A.2) (omitting the invariant terms) and (A.3) as in the AIC (Akaike 1974), we obtain the AIC-based partition criterion in Equation (11).